Fine's broadness conjecture for 01-polytopes and their polars

Let $P_{01,d}$ be the set of all $d$-dimensional $01$-polytopes, together with their polars. A set $P$ of $d$-polytopes is broad when every word-set $bb_s$ that is nonnegative on every polytope in $P$ is nonnegative on every $d$-polytope. Fine's broadness conjecture. The set $P_{01,d}$ is broad. If true, this gives an explicit finite family for the finite calculation that determines the extended $g$-vector in dimension $d$; the paper notes that there is little evidence beyond the satisfactory output of the $d=5$ computation.